On Thursday, as the Dow fell more than 300 points, Prof. Shiller told me, "The market has gone up for five years now and has gotten quite high, but I'm not selling yet." He advises investors to monitor not just the level of the market, but the "stories that people tell" about the market. If a sudden consensus about economic stagnation forms, that could be a dangerous "turning point," he says.

Shiller, who authored in the classic book on investing "Irrational Exuberance" and developed the CAPE, or cyclically-adjusted price-to-earnings, ratio measuring stock market valuation, told Zweig that while the CAPE ratio is elevated relative to history, "how do we know history hasn't changed?"

As of Friday's close, the CAPE ratio was at 25; the long-term average is about 16.

Late last month, Shiller appeared on CNBC and floated the idea that perhaps the stock market is going higher because investors are nervous.

"You might think this should be the reverse, that the stock market should go down when people are anxious. People are anxious because of (geopolitical unrest), because of inequality, and people are worried about being replaced by a computer. All these things are on people's minds."

But because of these anxieties, Shiller said, people want to save more, and because the economy has been weak, investors bid up the price of everything, and the stock market has been a major beneficiary of that.

"A bubble is a product of feedback from positive price changes that create a 'new era' ambience in which people think increasingly that prices will go up forever," Shiller told Zweig, adding that the bond market now "is just the opposite of a new-era ambience."

Shiller told Zweig that instead, the demand for bonds is driven by "an underlying angst" about the future of the global economy.